Problem 62 - Introductory Algebra for College Students

Question

Determine whether statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\frac{\frac{1}{4}-\frac{1}{3}}{\frac{1}{3}+\frac{1}{6}}=\frac{1}{12} \div \frac{3}{6}=\frac{1}{6}\)

Step-by-Step Solution

Answer

The given statement is false; the corrected true statement should be: \(\frac{\frac{1}{4}-\frac{1}{3}}{\frac{1}{3}+\frac{1}{6}}=\frac{1}{12} \div \frac{1}{1}=\frac{1}{12}.\)

Step 1: Examine the problem and break it down

It's necessary to study and understand the problem. In this case, it's an equation that involves operations with fractions. The equation is read as follows: \(\frac{\frac{1}{4}-\frac{1}{3}}{\frac{1}{3}+\frac{1}{6}}=\frac{1}{12} \div \frac{3}{6}.\) It must be simplified and checked if it equals \(\frac{1}{6}.\)

Step 2: Simplify the equation

The left side of the equation can be simplified to \(\frac{1}{12},\) and the right side can be simplified to \(\frac{1}{2},\) which clearly shows a discrepancy.

Step 3: Compare and analyze your results

The left side of the equation should match the right side if the statement is true. However, \(\frac{1}{12}\) which is the result of the left side does not equal \(\frac{1}{2},\) which is the result of the right side. Therefore, the statement as given is false.

Step 4: Provide the correct equation

To make the statement true, the equation on the right side should be changed so that \( \frac{1}{12} \div \frac{1}{1}\) which will simplify to \(\frac{1}{12},\) matching the left side.

Key Concepts

Simplifying Fractions

Understanding how to simplify fractions is crucial to mastering the basics of arithmetic. Simplification involves reducing a fraction to its smallest or simplest form, where the numerator and denominator have no common factors other than 1. For example, if we have the fraction \(\frac{8}{10}\), we notice that both 8 and 10 are divisible by 2. Dividing both by 2, we get \(\frac{4}{5}\), which is the simplified form of the original fraction.

In mathematics, we often simplify fractions to make calculations easier and results clearer to interpret. When working with complex equations involving fractions, simplifying at each step can prevent errors and make it easier to compare results, as seen in the exercise we're discussing. Simplifying the left side of the equation \(\frac{1}{4}-\frac{1}{3}\) gives us \(\frac{1}{12}\), which is already in its simplest form with no common factors between the numerator and the denominator.

Comparing Fractions

To determine if two fractions are equivalent, we must compare them. Comparing fractions usually involves ensuring they have a common denominator or converting them into decimal form. In the context of our exercise, we have the need to compare the simplified left side \(\frac{1}{12}\) with the right side after performing the division \(\frac{1}{2}\).

It's clear that \(\frac{1}{12}\) is not the same as \(\frac{1}{2}\), which indicates a discrepancy between the two sides of the equation. In more straightforward cases, we can visually see that 1/12 is much smaller than 1/2 without further calculations. When the fractions are less obvious, we could find a common denominator or convert to decimal to compare them directly. For example, \(\frac{1}{12} \approx 0.083\) and \(\frac{1}{2} = 0.5\). Here, we can easily see that 0.083 is much smaller than 0.5, confirming that the original equation was false.

Algebraic Expressions

Working with algebraic expressions is a step beyond basic arithmetic, involving not just numbers but also variables. However, the principles of operating with fractions still apply when handling algebraic expressions. An algebraic expression is a combination of constants, variables, and operations such as addition, subtraction, multiplication, and division.

For example, expression \(\frac{2x}{3y} - \frac{4y}{3x}\) involves both variables and operation with fractions. Simplifying algebraic expressions requires careful handling of both numerical coefficients and variables. In an equation, it's equally important to maintain equality by performing the same operations on both sides. This concept is pivotal in understanding and solving more complex problems where variables are involved, much like the fraction operations in our given exercise.